Tag Archives: Symmetry

The Dalí Theatre-Museum, opened in 1974, is the largest surrealistic object in the World. It was built on the ruins of the ancient theater of Figueres and hosts the most important collection of Dalí’s pictures and sculptures.

SalvadorFelipeJacintoDalí iDomènech,Marquis ofPúbol (11 May1904 – 23 January 1989)was born in Figueres. Although his principal mean of expression was the painting, he also made inroads in different fields such as cinema, photography, sculpture, fashion, jewellery and theatre, in collaboration with a wide range of artists in different media. His wife and muse, Gala Dalí was one of the essential characters in his biography. His public appearances never failed to impress and his ambiguous relationship with Francisco Franco’s regime made of this multifaceted character an icon of the 20th century and more than an artist. During his life he lived in Madrid, Paris and Catalonia and for this reason he was influenced by other important artists. He died in Barcelona and was buried in his own museum against his desire.

Dalí’s tomb. Source: Wikimedia Commons

Why did I say that he is more than an artist? If you visit the Dalí’s Theatre-Museum in Figueres, you will see his art based on mathematics and physical laws. Dalí’s relationship with science began in his teens when he started reading scientific articles and this passion for science was preserved all his life. In the museum you can find a great reflection of that passion. Furthermore, the painter’s library contains hundreds of books with notes about various scientific topics: physics, quantum mechanics, life’s origin, evolution and mathematics. In addition to that, he was subscribed to several scientific journals to be informed about the new scientific advances.

To show this relation between Mathematics and his masterpieces, I will explain three artworks which are exhibited in the museum from a mathematical point of view. The first one is Leda Atomica (1949). He created it from studying Luca Pacioli’s De Divina Proportione (Milan, 1509) Dalí made different computations for three months with the help of Matila Ghyka (1881-1965). Ghyka wrote some mathematical treatises related with the golden number like Le nombre d’or: Rites et rythmes pythagoriciens dans le development de la civilisation occidentale (1931), The Geometry of Art and Life (1946) or A Practical Handbook of Geometry and Design (1952).

Matila GhykaSource: Wikimedia Commons

The painting synthesizes centuries of tradition of Pythagorean symbolic Mathematics. It is a watermark based on the golden ratio, but making the viewer not appreciate it at first glance. In 1947’s sketch, it can be noticed the geometric accuracy of the analysis done by Dalí based on the Pythagorean mystic staff, which is a five-pointed star drawn with five straight strokes:

Source: Wikimedia Commons

You can see that Gala, in the centre of the painting, is enclosed in a regular pentagon and her proportions are according the golden ratio. The picture depicts Leda, the mythological queen of Sparta, with a swan suspended behind her left. There also are a book, a set square, two stepping tools and a floating egg. Dalí himself described the picture in the following way:

Dalí shows us the hierarchized libidinous emotion, suspended and as though hanging in midair, in accordance with the modern ‘nothing touches’ theory of intra-atomic physics. Leda does not touch the swan; Leda does not touch the pedestal; the pedestal does not touch the base; the base does not touch the sea; the sea does not touch the shore…

Leda atomica (1949)( Source: Wikimedia Commons

Another mathematical example is Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors from 1973. This is a stereoscopic work which is an example of the experiments conducted by him during the seventies. Dalí wished to reach the third dimension through stereoscopy and to achieve the effect of depth.

Dalí from the Back Painting Gala from the Back Eternalised by Six Virtual Corneas Provisionally Reflected in Six Real Mirrors (1973)Photography by Roger Pijoan Català

The last example is Nude Gala Looking at the Sea Which at 18 Meters Appears the President Lincoln (1975). In this case, Dalí used the double image techinque for creating akind of illusion which is very common in his work.

Nude Gala Looking at the Sea Which at 18 Meters Appears the President LincolnPhotography by Roger Pijoan Català

So, Dalí was more mathematician than one can imagine.

This post has been written by Sara Puig Cabruja in the subject Història de les Matemàtiques (History of Mathematics, 2014-15).

Aljaferia Palace is one of the most beautiful Islamic palaces which can be visited in Spain. It was built in the second half of the 11th century in the Moorish taifa os Saraqusta (present day Zaragoza) by the King al-Muqtâdir Bânû Hûd.

Source: Wikimedia Commons

I’m sure that you are wondering why I am talking about this building now. The building is wonderful but this is not the reason. Do you know who King al-Mu’tamân is? No? King al-Mu’tamân (1081-1085) grew in this palace and was educated under teachers and philosphers. Before 1081, he began to write an encyclopaedic work about Mathematics (Kitâb al-Istikmâl or Book of the Perfection) with his collaborators’ contributions. Al-Mu’tamân wanted to write the most important mathematical treatise until that time. Only four hundred propositions about Classic Geometry have survived: some results from Euclid’s Elements and Data, Apollonius’ Conics, Archimedes’ On the sphere and the cylinder, Theodosius’ Spherics, Menalaus’ Spherics and Ptolemy’s Almagest. There also are Arabic contributions as Thâbit b. Qurra’s treatise on amicable numbers, some of the Bânû Mûsâ’s works, Ibrâhim b. Sinân’s The Quadrature of the Parabola and Ibn al-Haytham’s Optics, On the Analysis and the Synthesis and On the given things. One of the most interesting results is the demonstrarion of Ceva’s Theorem (attributed to the Italian mathematician Giovanni Ceva (d. 1734) ). Unfortunately, al-Mu’tamân became King of Saraqusta in 1081 and the Book of Perfection was never finished so the sections about Astronomy and Optics weren’t writen. The Book of Perfection was commented by Maimonides (1135-1204) some years later.

Photography by Carlos Dorce

In 1118 King Alfonso I of Aragon conquered Zaragoza and after a lot of years, the palace became the royal residence. Nowadays, we can visit most of its rooms included Catholic Monarchs‘s throne room. Can you imagine young al-Mu’tamân playing with his friends in this idilic place?

Photography by Carlos Dorce

Or praying in the octogonal Oratory?

Photography by Carlos Dorce

Visiting the Palace, we can see a very good quotation about the importance of the Geometry in the Islamic art:

The preference of the Islamic culture for abstract art developed a type of decoration based on geometric order, its main argument being repeated themes and the objective of suggesting infinity. Of great importance in this concept was the development of mathematics in the Muslim civilization, which were then skillfull applied to construction and decoration. Starting off with a few examples of symmetry, Hispano-Muslim and then Mudejar art was capable of developing complex decorative themes that were always based on repetition.

The forntispiece of Museo del Prado of Madrid is full of allegorical figures of the muses and the arts. If we watch it carefully, we’ll notice Urania with a compass and a globe in her hands counting on a parchment:

Photography by Carlos Dorce

The building was designed by the architect Juan de Villanueva (1739-1811) and it had to host the Royal Observatory, a Science Room, the Botanic Gardens, schools, laboratories,… The Spanish king yhought that it could be a very good example of the new illustrated Spain. However, it never was used in this way:

Photography by Carlos Dorce

Nowadays thousands of tourists visit the pictures in the Museo del Prado and only a few ones visit outside the building. Among all the statues which decorate this neoclassical structure there are the Architecture…

Photography by Carlos Dorce

…and the Symmetry:

Photography by Carlos Dorce

There are also some medallions with busts of famous Spanish scientist and writers on each of these statues. Of course, Juan de Herrera is also here:

Segovia is one of the most beautiful Spanish cities. The staple of the city is the Aqueduct (Ist century AD) located in the Plaza del Azoguejo but all the old city is a monument which must be visited if you come to Spain. Another emblematic building of the city is the Alcazar (XIInd century) which was the Royal palace of the Kings of Castile in Medieval times:

Alcazar of SegoviaPhotography by Carlos Dorce

Segovia is also a very mathemtical place. There is not any particular museum or palace but a walk around the old city displays the reason why I am writing this post: it is full of mosaics on the facades of the buildings:

Photography by Carlos Dorce

There are a lot of buildings with this kind of decoration and we can find some examples of the 17 symmetry groups in which we can clasify all the mosaics represented on the plane. For example:

Photography by Carlos Dorce

Cascales PalacePhotography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

Photography by Carlos Dorce

If you want to take pictures of mosaics you must go to Segovia and enjoy its mathematical facades. I think that I must come back to Segovia to o a mathematical study about all these wonderful mosaics.

Josef Albers was a German-born American abstract artist whose study of color made possible to create some beautiful geometrical compositions as the Homage to the Square: Apparition (1959) kept in the Guggenheim Museum of New York. He enrolled at the Bauhaus school of art in Weimar and where was appointed as a teacher in 1925. He remained in the school until the school closed in 1933 and then he emigrated to the USA. In 1950 he joined the University of Yale and began his famous Homage to the Square series. Here you have two paintings more:

Homage to the Square: With Rays (1959) Metropolitan Museum of Art of New York

Homage to the Square: Soft Spoken (1969)Metropolitan Museum of Art of New York

Kenneth Snelson (born June 29, 1927) is a contemporary sculptor who arranges rigid and flexible components to compose his sculptures combining tension and structural integrity. This Neddle Tower II (1969) is 30 meters high and it’s interesting here because of this picture:

Source: Wikimedia Commons

Is a mathematical picture or not? The sculpture is in the garden of the Kröller-Müller Museum in Otterlo.

Snelson discovered the underlying principle for such structures in 1948, advocating the term “floating compression” to describe the balance between tension and compression and, in his sculptures, between flexible cables and rigid tubes. R. Buckminster Fuller (1895-1983) coined the word “tensegrity” (combining “tension” and “integrity”) for the same idea, and his term stuck. Snelson refers to weaving as the “mother of tensegrity.”

Snelson defines “tensegrity” as follows: “Tensegrity describes a closed structural system composed of a set of three or more elongate compression struts within a network of tension tendons, the combined parts mutually supportive in such a way that the struts do not touch one another, but press outwardly against nodal points in the tension network to form a firm, triangulated, prestressed, tension and compression unit.”

Snelson’s Needle Tower delivers a wonderful geometrical surprise when you venture underneath and look up to see a striking pattern of six-pointed stars.

This pattern arises naturally out of the requirement that each layer of a tensegrity structure consist of three compression elements (tubes). The sets of three alternate, giving the impression of a six-pointed star as you look up the tower. Snelson’s sculptures often show this kind of symmetry.

The elegance of Snelson’s tower suggests its use as an aesthetic alternative to conventional communications towers. But tensegrity structures are fairly elastic and flexible. They sway in the wind, which may not be ideal for the antennas and dishes that would top such structures.